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PTAS for k-tour cover problem on the plane for moderately large values of k

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Adamaszek, Anna, Czumaj, Artur and Lingas, Andrzej. (2010) PTAS for k-tour cover problem on the plane for moderately large values of k. International Journal of Foundations of Computer Science, Volume 21 (Number 6). pp. 893-904. ISSN 0129-0541

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Official URL: http://dx.doi.org/10.1142/S0129054110007623

Abstract

Let P be a set of n points in the Euclidean plane and let O be the origin point in the plane. In the k-tour cover problem (called frequently the capacitated vehicle routing problem), the goal is to minimize the total length of tours that cover all points in P, such that each tour starts and ends in O and covers at most k points from P.

The k-tour cover problem is known to be NP-hard. It is also known to admit constant factor approximation algorithms for all values of k and even a polynomial-time approximation scheme (PTAS) for small values of k, k = circle divide (log n/ log log n).

In this paper, we significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of k <= 2(log delta n), where delta = delta(epsilon). The main technical result proved in the paper is a novel reduction of the k-tour cover problem with a set of n points to a small set of instances of the problem, each with circle divide((k/epsilon)(circle divide(1))) points.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics
Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software

Divisions:

Faculty of Science > Computer Science

Library of Congress Subject Headings (LCSH):

Transportation problems (Programming), Approximation algorithms

Journal or Publication Title:

International Journal of Foundations of Computer Science

Publisher:

World Scientific Publishing Co. Pte. Ltd.

ISSN:

0129-0541

Official Date:

December 2010

Dates:

Date	Event
December 2010	Published

Volume:

Volume 21

Number:

Number 6

Page Range:

pp. 893-904

Identification Number:

10.1142/S0129054110007623

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Restricted or Subscription Access

Funder:

Engineering and Physical Sciences Research Council (EPSRC), Sweden. Vetenskapsrådet [Research Council], University of Warwick. Centre for Discrete Mathematics and Its Applications

Grant number:

EP/D063191/1 (EPSRC), 621-2005-408 (VR)

Version or Related Resource:

Adamaszek, Anna, et al. (2009). PTAS for k-tour cover problem on the plane for moderately large values of k. Lecture Notes in Computer Science, 5878, pp. 994-1003. http://wrap.warwick.ac.uk/id/eprint/5481

Related URLs:

Related item in WRAP

References:

[1] S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman
and other geometric problems. Journal of the ACM, 45(5):753{782, 1998.
[2] T. Asano, N. Katoh, H. Tamaki, and T. Tokuyama. Covering points in the plane by
k-tours: a polynomial time approximation scheme for �xed k. IBM Tokyo Research
Laboratory Research Report RT0162, 1996.
[3] T. Asano, N. Katoh, H. Tamaki, and T. Tokuyama. Covering points in the plane by
k-tours: Towards a polynomial time approximation scheme for general k. Proceedings
of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 275{283,
1997.
[4] B.S. Baker. Approximation algorithms for NP-complete problems on planar graphs.
Journal of the ACM, 41(1):153{180, 1994.
[5] G. B. Dantzig and R, H. Ramser. The truck dispatching problem. Management Sci-
ence, 6(1):80{91, October 1959.
[6] A. Das and C. Mathieu. A quasi-polynomial time approximation scheme for Euclidean
capacitated vehicle routing. Proceedings of the 21st Annual ACM-SIAM Symposium
on Discrete Algorithms (SODA), pp. 390{403, 2010.
[7] M. L. Fischer. Vehicle routing. In Network Routing, Handbooks in Operations Re-
search and Management Science, Vol. 8, edited by M. O. Ball, T. L. Magnanti,
C. L. Monma, and G. L. Nemhauser, Elsevier, Amsterdam, pp. 1{33, 1995.
[8] M.R. Garey and D.S. Johnson. Computers and Intractability. A Guide to the Theory
of NP-completeness. W.H. Freeman and Company, New York 1979. [9] B. Golden, S. Raghavan, and E.Wasil. The Vehicle Routing Problem: Latest Advances
and New Challenges. Springer, New York, 2008.
[10] M. Haimovich and A.H.G. Rinnooy Kan. Bounds and heuristics for capacitated rout-
ing problems. Mathematics of Operation Research, 10(4):527{542, 1985.
[11] G. Laporte. The vehicle routing problem: An overview of exact and approximate
algorithms. European Journal of Operational Research, 59(3):345{358, 1992.
[12] J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A sim-
ple polynomial-time approximation scheme for geometric TSP, k-MST, and related
problems. SIAM Journal on Computing, 28(4):1298{1309, August 1999.
[13] F. Preparata and M. Shamos. Computational Geometry { an Introduction. Springer
Verlag, New York, NY, 1985.
[14] P. Toth and D. Vigo. The Vehicle Routing Problem. SIAM, Philadelphia, 2001.